Chapter 6 Polar Duality, Polyhedra and Polytopes 6.1 Polarity and Duality In this section, we apply the intrinsic duality afforded by a Euclidean structure to the study of convex sets and, in particular, polytopes. Let E = En be a Euclidean space of dimension n. Pick any origin, O,inEn (we may assume O =(0,...,0)). We know that the inner product on E = En induces a ∗ duality between E and its dual E , namely, u → ϕu, where ϕu is the linear form defined by ϕu(v)=u · v,for all v ∈ E. 205 206 CHAPTER 6. POLAR DUALITY, POLYHEDRA AND POLYTOPES For geometric purposes, it is more convenient to recast this duality as a correspondence between points and hy- perplanes, using the notion of polarity with respect to the unit sphere, Sn−1 = {a ∈ En |Oa =1}. First, we need the following simple fact: For every hy- perplane, H, not passing through O, there is a unique point, h, so that H = {a ∈ En | Oh · Oa =1}. Using the above, we make the following definition: Definition 6.1.1 Given any point, a = O,thepolar hyperplane of a (w.r.t. Sn−1) or dual of a is the hyper- plane, a†, given by a† = {b ∈ En | Oa · Ob =1}. Given a hyperplane, H, not containing O,thepole of H (w.r.t Sn−1)ordual of H is the (unique) point, H†,so that H = {a ∈ En | OH† · Oa =1}. 6.1. POLARITY AND DUALITY 207 We often abbreviate polar hyperplane to polar. We immediately check that a†† = a and H†† = H,so, we obtain a bijective correspondence between En −{O} and the set of hyperplanes not passing through O. When a is outside the sphere Sn−1, there is a nice geo- metric interpetation for the polar hyperplane, H = a†. Indeed, in this case, since H = a† = {b ∈ En | Oa · Ob =1} and Oa > 1, the hyperplane H intersects Sn−1 (along an (n − 2)-dimensional sphere) and if b is any point on H ∩ Sn−1, we claim that Ob and ba are orthogonal. This means that H ∩ Sn−1 is the set of points on Sn−1 where the lines through a and tangent to Sn−1 touch Sn−1 (they form a cone tangent to Sn−1 with apex a). 208 CHAPTER 6. POLAR DUALITY, POLYHEDRA AND POLYTOPES b O a a† Figure 6.1: The polar, a†,ofapoint,a, outside the sphere Sn−1 Also, observe that for any point, a = O, and any hy- perplane, H, not passing through O,ifa ∈ H, then, H† ∈ a†, i.e, the pole, H†,ofH belongs to the polar, a†, of a. If a =(a1,...,an), the equation of the polar hyperplane, a†,is a1X1 + ···+ anXn =1. Now, we would like to extend this correspondence to sub- sets of En, in particular, to convex sets. 6.1. POLARITY AND DUALITY 209 Given a hyperplane, H, not containing O,wedenoteby H− the closed half-space containing O. Definition 6.1.2 Given any subset, A,ofEn, the set ∗ n † A = {b ∈ E | Oa·Ob ≤ 1, for all a ∈ A} = (a )−, a∈A a= O is called the polar dual or reciprocal of A. † † To simplify notation we write a− for (a )−. Note that ∗ n † n {O} = E ,soitisconvenienttosetO− = E ,even though O† is undefined. † † We use a different notation, a and H , for polar hy- perplanes and poles, as opposed to A∗, for polar duals, to avoid confusion. Indeed, H† and H∗, where H is a hyperplane (resp. a† and {a}∗, where a isapoint)are very different things! 210 CHAPTER 6. POLAR DUALITY, POLYHEDRA AND POLYTOPES v3 v2 v4 v1 v5 Figure 6.2: The polar dual of a polygon In Figure 6.2, the polar dual of the polygon (v1,v2,v3,v4,v5) is the polygon shown in green. This polygon is cut out by the half-planes determined by the polars of the vertices (v1,v2,v3,v4,v5) and containing the center of the circle. By definition, A∗ is convex even if A is not. 6.1. POLARITY AND DUALITY 211 Furthermore, note that (1) A ⊆ A∗∗. (2) If A ⊆ B,thenB∗ ⊆ A∗. (3) If A is convex and closed, then A∗ =(∂A)∗. It follows immediately from (1) and (2) that A∗∗∗ = A∗. Also, if Bn(r) is the (closed) ball of radius r>0 and cen- ter O, it is obvious by definition that Bn(r)∗ = Bn(1/r). We would like to investigate the duality induced by the operation A → A∗. Unfortunately, it is not always the case that A∗∗ = A, but this is true when A is closed and convex, as shown in the following proposition: 212 CHAPTER 6. POLAR DUALITY, POLYHEDRA AND POLYTOPES Proposition 6.1.3 Let A be any subset of En (with origin O). ◦ ◦ (i) If A is bounded, then O ∈ A∗;ifO ∈ A,thenA∗ is bounded. (ii) If A is a closed and convex subset containing O, then A∗∗ = A. Note that A∗∗ = {c ∈ En | Od · Oc ≤ 1 for all d ∈ A∗} = {c ∈ En | (∀d ∈ En)(if Od · Oa ≤ 1 for all a ∈ A, then Od · Oc ≤ 1)}. 6.1. POLARITY AND DUALITY 213 Remark: For an arbitrary subset, A ⊆ En,itcanbe shown that A∗∗ = conv(A ∪{O}), the topological clo- sure of the convex hull of A ∪{O}. Proposition 6.1.3 will play a key role in studying poly- topes, but before doing this, we need one more proposi- tion. Proposition 6.1.4 Let A be any closed convex sub- ◦ set of En such that O ∈ A. The polar hyperplanes of the points of the boundary of A constitute the set of supporting hyperplanes of A∗. Furthermore, for any a ∈ ∂A, the points of A∗ where H = a† is a sup- porting hyperplane of A∗ are the poles of supporting hyperplanes of A at a. 214 CHAPTER 6. POLAR DUALITY, POLYHEDRA AND POLYTOPES 6.2 Polyhedra, H-Polytopes and V-Polytopes There are two natural ways to define a convex polyhedron, A: (1) As the convex hull of a finite set of points. (2) As a subset of En cut out by a finite number of hyper- planes, more precisely, as the intersection of a finite number of (closed) half-spaces. As stated, these two definitions are not equivalent because (1) implies that a polyhedron is bounded, whereas (2) allows unbounded subsets. Now, if we require in (2) that the convex set A is bounded, it is quite clear for n = 2 that the two definitions (1) and (2) are equivalent; for n = 3, it is intuitively clear that definitions (1) and (2) are still equivalent, but proving this equivalence rigorously does not appear to be that easy. What about the equivalence when n ≥ 4? 6.2. POLYHEDRA, H-POLYTOPES AND V-POLYTOPES 215 It turns out that definitions (1) and (2) are equivalent for all n, but this is a nontrivial theorem and a rigorous proof does not come by so cheaply. Fortunately, since we have Krein and Milman’s theorem at our disposal and polar duality, we can give a rather short proof. The hard direction of the equivalence consists in proving that definition (1) implies definition (2). This is where the duality induced by polarity becomes handy, especially, the fact that A∗∗ = A! (under the right hypotheses). First, we give precise definitions (following Ziegler [?]). 216 CHAPTER 6. POLAR DUALITY, POLYHEDRA AND POLYTOPES (a) (b) Figure 6.3: (a) An H-polyhedron. (b) A V-polytope Definition 6.2.1 Let E be any affine Euclidean space 1 of finite dimension, n. An H-polyhedron in E, for short, P p C E a polyhedron, is any subset, = i=1 i,of defined as the intersection of a finite number, p ≥ 1, of closed half- spaces, Ci;anH-polytope in E is a bounded polyhedron and a V-polytope is the convex hull, P =conv(S), of a finite set of points, S ⊆E. Examples of an H-polyhedron and of a V-polytope are shown in Figure 6.3. −→ 1This means that the vector space, E , associated with E is a Euclidean space. 6.2. POLYHEDRA, H-POLYTOPES AND V-POLYTOPES 217 Obviously, polyhedra and polytopes are convex and closed (in E). Since the notions of H-polytope and V-polytope are equivalent (see Theorem 6.3.1), we often use the sim- pler locution polytope. Note that Definition 6.2.1 allows H-polytopes and V- polytopes to have an empty interior, which is sometimes an inconvenience. This is not a problem. In fact, we can prove that we may always assume to E = En and restrict ourselves to the affine hull of P (some copy of Ed,ford ≤ n, where d = dim(P ), as in Definition 3.1.1). Proposition 6.2.2 Let A ⊆E be a V-polytope or an H-polyhedron, let E =aff(A) be the affine hull of A in E (with the Euclidean structure on E induced by the Euclidean structure on E)andwrited = dim(E). Then, the following assertions hold: (1) The set, A,isaV-polytope in E (i.e., viewed as a subset of E)iffA is a V-polytope in E. (2) The set, A,isanH-polyhedron in E (i.e., viewed as a subset of E)iffA is an H-polyhedron in E. 218 CHAPTER 6. POLAR DUALITY, POLYHEDRA AND POLYTOPES The following simple proposition shows that we may as- sume that E = En: Proposition 6.2.3 Given any two affine Euclidean spaces, E and F ,ifh: E → F is any affine map then: (1) If A is any V-polytope in E,thenh(E) is a V- polytope in F .